Let &pi;&colon;E&srarr;&period;Mbe a fiber bundle, with base dimension nand fiber dimension m and let &pi;&ell;&colon;J&ell;E&srarr;Mbe the &ell;-th jet bundle. The Prolong command will take a geometry object defined, either on E or on J&ell;E&comma;and extend or lift that object to a higher order jet space J&ell;E. The lifting or prolongation procedures considered here require only algebraic operations and differentiations. There are 4 different types of prolongation which can be performed by the command Prolong.

1. Prolongation of Jet Spaces. Suppose that the command DGsetup has been used to initialize a jet space J&ell;E. This means that the standard jet space coordinates &lpar;xi&comma;u&alpha;&comma;ui&alpha;&comma;uij&alpha;, ..., uij⋅⋅⋅&ell;&alpha;&rpar;are protected. The coordinate vector fields, coordinate 1-forms, and contact forms to order &ell;are initialized and protected. The command Prolong(k), where k&GreaterEqual;l&comma;with extend these protections and definitions to order k. The result is same as making a call to DGsetup to initialize the jet space JkE&comma;but is slightly faster since Prolong command only needs to define and protect the coordinates,vectors and 1 -forms from order &ell;&plus;1to k&period;

2. Prolongation of Vector Fields. Let Z be a vector field on JkE&period;We say that Zpreserves the contact ideal onJkE if for any contact form &Theta;&comma;the Lie derivative &Laplacetrf;Z&Theta;is also a contact form. Let Xbe a projectable, point, contact, evolutionary, total,or generalized vector field with values in the tangent space E.(See AssignVectorType for the definitions of these types of vector fields.) Then, for each k, there is a unique vector field Zon JkEwhich preserves the contact ideal on JkEand which projects pointwise to X&period; This vector field Z is called the prolongation of X to order k. and is denoted by prkX. The explicit formula for vector field prolongation is given below. The second calling sequence Prolong(X, k) computes the prolongation of the vector field Xto order k&period;

3. Prolongation of Transformations. Let E&srarr;Mand F&srarr;Nbe two fiber bundles. We say that a transformation &psi;&colon;J&ell;E&srarr;JnFis a generalized contact transformation if for every contact form &Theta; on JnF, the pullback &psi;&ast;&Theta;is a contact form on J&ell;E. Let &phi; be a projectable transformation, a point transformation, a contact transformation, a differential substitution or a generalized differential substitution. These maps are defined as mappings from JpEtoJqFfor the appropriate values of p&comma;q&period;(See AssignTransformationType for the definitions of these different types of transformations.) Then, for each k, there is a unique generalized contact transformation ψ&colon;Jp&plus;kE&srarr;Jq&plus;kFwhich covers &phi;. This transformation &Psi;is called the prolongation of &phi;to orderk and it denoted by prk&phi;.The third calling sequence Prolong(&phi;&comma;k) computes the prolongation of &phi;to order k&period;

4. Prolongation of Differential Equations. A system of &ell;-th order differential equations can defined as the zero set of a collection &Delta; of functions Fa&colon;J&ell;E&srarr;&reals;.The k−thorder prolongation of &Delta;&comma;denote by prk&Delta;is the system of (&ell; +k)-th order differential equations defined as the zero set of the functionsFaand all their total derivatives Di1Di2⋅⋅⋅DitFa to order t&leq;k. The fourth calling sequence Prolong(Delta, k) computes the prolongation of a system of differential equations &Delta;to order k.Use the command DifferentialEquationData to convert a list of functionsFa&colon;J&ell;E&srarr;&reals;into a differential equation data structure that can be passed to the Prolong command. The result is a new differential equation data structure representing the prolongation of the differential equations.

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If a vector field, transformation or differential equation has been prolonged to a certain order using Prolong,then the prolonged objects may themselves be prolonged to a higher order using Prolong.

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The command Prolong is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form Prolong(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-Prolong(...).

Details

If X&equals;Aj&PartialD;&PartialD;xj&plus;B&alpha;&PartialD;&PartialD;u&alpha;is a generalized vector field on E, then the k-th prolongation of X is the vector field

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